Small scratch - car insurance

[Notsurehowtofeel2025]

Our 9yo child was getting a lift in a friend's car and when he opened his door, he scratched the car parked next to them.
Theres no damage to friend's car and what looks like a small scratch (on a 10yo car) on the other car.
They swapped numbers and when friend let us know about it we obviously offered to pay for repairs to the other car.
The other car is saying they'd rather go through insurance which is legally their right so not much we can do.
My question is, what will happen with both policies / excesses etc?
I don't want our friend to be financially affected for our child's accident.
Will the other car drivers insurance not also go up? Struggling to understand why they want to go through insurance to be honest.

It can be more difficult to go through insurance if you don't do it immediately, so if you agree to pay but then won't transfer them the money. So I wouldn't want to go through insurance, but it might explain why they're hesitant.

It's 3 local families involved and we all know where each other lives so I don't think they'd be worried that we wouldn't pay but maybe they've been stung in the past.
do you know what will happen in terms of excess etc? Looking online it seems there's no excess to pay but I don't understand how that can be. Do premium then go up?

I would make it clear that if they can get a quote you’d send cash, straight away. Some people just don’t realise that regardless of fault these things cause extra premiums for all involved, and need it spelt out

We have honestly made it really clear. I think they're just being difficult tbh. At this point I'm just worried about my friends premiums going up. A bit embarrassing to cost friends money because my child has been careless